Oh boy, this is actually a mathematical question, look up median, modal, average, etc in google and enjoy the maths. The average firm means whatever you want it to mean. The medium sized firm can mean a lot of things but probably means take the smallest firm (1 person) and the largest firm (say 100,000) people and then the medium has 50,000 people. So many ways to calculate these figures.

Not really. It can, but it does not necessarily mean the same thing. Medium, means not small and not large. Average means the most common. So, it can be the same thing, but it doesn't have to be.

In U.K. english "average " can also by a bit of an insult. For example if you were asked to describe how good looking someone was and you said "average " well that is a bit insulting really. If some one said you were just average you would not be too pleased!

Yes and no. In general, "medium" is a vague word meaning something that is neither "small" nor "large," but somewhere in between. "Average" literally is a technical term in mathematics and statistics. It means a calculated number that is one of several ways of measuring the central or middle value of a group of values. For example, "the average height of a man in the U.S. is 177 cm." "Average" also has the everyday meaning of common, ordinary, or normal. In the case of a business, you could say that a "medium-sized" firm means "not a big firm and not a small firm," while an "average-sized firm" means "the size that most firms usually are." In this case, both are imprecise expressions and really mean the same thing. An "average firm" doesn't necessarily mean an average-SIZED firm. It means a firm that is normal, typical, not unusual, in some or most respects. For example, one might say--I think this is true, but I'm not sure!--"in the United States, the average firm would have its payroll handled by an outside payroll service."